Section 33: The Urysohn Lemma

1. Two subsets $A,B\subseteq X$ are said to be separated by a continuous function if there is a continuous function $f:X\to[0,1]$ such that $f(A)=\{0\}$ and $f(B)=\{1\}$ .
2. Urysohn Lemma: if $X$ is normal and $A,B\subseteq X$ are closed and disjoint, then they can be separated by a continuous function.
3. Consider $U_p=\{x:f(x)<p\}$ and $U_p'=\{x:f(x)\le p\}$ : the first is open and contained in the second which is closed, moreover, for every $q>p$ : $U_p'\subseteq U_q$ . We use the normality of the space to construct a sequence of such open sets and define $f(x)=\inf\{p:x\in U_p\}$ .

Why cannot the proof of the Urysohn lemma be generalized to show that in a regular space... you can also separate points from closed sets by continuous functions? At first glance, it seems that the proof of the Urysohn lemma should go through... Requiring that one be able to separate a point from a closed set by a continuous function is, in fact, a stronger condition...

1. Completely regular space: each one point set is closed and can be separated from a disjoint closed set by a continuous function.

In the early years of topology, the separation axioms... were labelled $T_1$ , $T_2$ (Hausdorff), $T_3$ (regular), $T_4$ (normal), and $T_5$ (completely normal)... The letter "T" stands for the German "Trennungsaxiom," which means "separation axiom." Later, when the notion of complete regularity was introduced, someone suggested facetiously that it should be called the "$T_{3\frac{1}{2}}$ axiom"... The terminology is in fact sometimes used in the literature!

1. Subspaces and complete regularity:
2. A subspace of a completely regular space is completely regular.
3. If $A,B$ are disjoint closed sets and one of them is compact, they can be separated by a continuous function.
4. Products and complete regularity:
5. The product of any family of completely regular spaces is completely regular.
6. A connected regular space with at least two points is uncountable.
7. But there is a countably infinite connected Hausdorff space.
8. Perfectly normal space: a normal space such that every closed set is a $G_\delta$ -set: the intersection of a countably many open sets.
9. $f:X\Rightarrow\mathbb{R}_+$ is said to vanish precisely on $A$ if $A=f^{-1}(\{0\})$ .
10. The Strong Urysohn Lemma: every two disjoint closed $G_\delta$ subsets $A$ and $B$ of a normal space $X$ can be separated by a continuous function $f:X\to[0,1]$ such that $f$ vanishes precisely on $A$ and $1-f$ vanishes precisely on $B$ .
11. $X$ is perfectly normal if and only if it is $T_1$ and for every closed set there is a continuous function that vanishes precisely on the set.
12. Perfectly normal implies completely normal.
13. Completely regular and perfectly normal spaces:
14. A topological group is completely regular.
15. A metric space is perfectly normal.